The arithmetic of quaternion algebras - Dartmouth College

304 Pages·2014·1.88 MB·English

The arithmetic of
quaternion algebras
JohnVoight
[email protected]
April21,2014

Preface
Goal
Intheresponsetoreceivingthe1996SteelePrizeforLifetimeAchievement[Ste96],
ShimuradescribesalecturegivenbyEichler:
[T]he fact that Eichler started with quaternion algebras determined his
course thereafter, which was vastly successful. In a lecture he gave in
Tokyohedrewahexagonontheblackboardandcalleditsverticesclock-
wise as follows: automorphic forms, modular forms, quadratic forms,
quaternionalgebras,Riemannsurfaces,andalgebraicfunctions.
This book is an attempt to fill in the hexagon sketched by Eichler and to augment
it further with the vertices and edges that represent the work of many algebraists,
geometers,andnumbertheoristsinthelast50years.
Quaternion algebras sit prominently at the intersection of many mathematical
subjects. Theycaptureessentialfeaturesofnoncommutativeringtheory,numberthe-
ory, K-theory, group theory, geometrictopology, Lietheory, functionsofacomplex
variable, spectral theoryofRiemannian manifolds, arithmeticgeometry, representa-
tion theory, the Langlands program—and the list goes on. Quaternion algebras are
especially fruitful to study because they often reflect some of the general aspects of
thesesubjects,whileatthesametimetheyremainamenabletoconcreteargumenta-
tion.
With this in mind, we have two goals in writing this text. First, we hope to in-
troduce a large subset of the above topics to graduate students interested in algebra,
geometry, and number theory. We assume that students have been exposed to some
algebraic number theory (e.g., quadratic fields), commutative algebra (e.g., module
theory, localization, and tensor products), as well as the basics of linear algebra,
topology,andcomplexanalysis. Forcertainsections,furtherexperiencewithobjects
in arithmetic geometry, such as elliptic curves, is useful; however, we have endeav-
oredtopresentthematerialinawaythatismotivatedandfullofrichinterconnections
i
ii
andexamples,sothatthereaderwillbeencouragedtoreviewanyprerequisiteswith
theseexamplesinmindandsolidifytheirunderstandinginthisway. Atthemoment,
one can find introductions for aspects of quaternion algebras taken individually, but
thereisnotextthatbringsthemtogetherinoneplaceandthatdrawstheconnections
between them; we have tried to fill this gap. Second, we have written this text for
researchers in these areas: we have collected results otherwise scattered in the liter-
ature, provide some clarifications and corrections and complete proofs in the hopes
thatthistextwillprovideaconvenientreferenceinthefuture.
In order to combine these features, we have opted for an organizational pattern
thatis“horizontal”ratherthan“vertical”: thetexthasmanychapters,eachrepresent-
ing a different slice of the theory. Each chapter could be used in a (long) seminar
afternoon or could fill a few hours of a semester course. To the extent possible,
we have tried to make the chapters stand on their own (with explicit references to
results used from previous chapters) so that they can be read based on the reader’s
interests—hopefullytheinterdependenceofthematerialwilldrawthereaderinmore
deeply! Theintroductorysectionofeachchaptercontainsmotivationandasummary
oftheresultscontainedtherein,andweoftenrestrictthelevelofgeneralityandmake
simplifying hypotheses so that the main ideas are made plain. Hopefully the reader
whoisnewtothesubjectwillfindthesehelpfulaswaytodivein.
This book has three other features. First, as is becoming more common these
days, paragraphs are numbered when they contain results that are referenced later
on;wehaveoptednottoputthesealwaysinalabelledenvironment(definition,theo-
rem,proof,etc.)tofacilitatetheexpositionalflowofideas,whileatthesametimewe
wishedtoremainpreciseaboutwhereandhowresultsareused. Second,wehavein-
cludedineachchapterasectionon“extensionsandfurtherreading”,wherewehave
indicated some of the ways in which the author’s (personal) choice of presentation
of the material naturally connects with the rest of the mathematical landscape. Our
general rule (except the historical expository in Chapter 1) has been to cite specific
results and proofs in the text where they occur, but to otherwise exercise restraint
until this final section where we give tangential remarks, more general results, ad-
ditional references, etc. Finally, in many chapters we have also included a section
on algorithmic aspects, for those who want to pursue the computational side of the
theory. And as usual, each section also contains a number of exercises at the end,
ranging from checking basic facts used in a proof to more difficult problems that
stretchthereader. Formanyoftheseexercises,therearehintsattheendofthebook;
foranyresultthatisusedlater,acompleteargumentisgiven.
iii
Acknowledgements
This book began as notes from a course offered at McGill University in the Winter
2010semester,entitledComputationalAspectsofQuaternionAlgebrasandShimura
Curves. IwouldliketothankthemembersofmyMath727classfortheirinvaluable
discussions and corrections: Dylan Attwell-Duval, Xander Faber, Luis Finotti, An-
drewFiori,CameronFranc,AdamLogan,MarcMasdeu,JungbaeNam,AurelPage,
Jim Parks, Victoria de Quehen, Rishikesh, Shahab Shahabi, and Luiz Takei. This
coursewaspartofthespecialthematicsemesterNumberTheoryasAppliedandEx-
perimentalScienceorganizedbyHenriDarmon,EyalGoren,AndrewGranville,and
Mike Rubinstein at the Centre de Recherche Mathe´matiques (CRM) in Montre´al,
Que´bec, and the extended visit was made possible by the generosity of Dominico
Grasso, dean of the College of Engineering and Mathematical Sciences, and Jim
Burgmeier, chairoftheDepartmentofMathematicsandStatistics, attheUniversity
ofVermont. Withgratitude,Iacknowledgetheirsupport.
ThewritingcontinuedwhiletheauthorwasonsabbaticalattheUniversityofCal-
ifornia, Berkeley, sponsored by Ken Ribet. Several students attended these lectures
and gave helpful feedback: Watson Ladd, Andrew Niles, Shelly Manber, Eugenia
Rosu, EmmanuelTsukerman, VictoriaWood,andAlexYoucis. Mysabbaticalfrom
DartmouthCollegefortheFall2013andWinter2014quarterswasmadepossibleby
the efforts of Associate Dean David Kotz, and I thank him for his support. Further
progressonthetextwasmadeinpreparationforaminicourseonBrandtmodulesas
partofMinicoursesonAlgebraicandExplicitMethodsinNumberTheory,organized
by Ce´cile Armana and Christophe Delaunay at the Laboratoire de Mathe´matiques
de Besanc¸on in Salins-les-Bains, France. Finally, I would like to thank the partici-
pants in my Math 125 Quaternion algebras class at Dartmouth in Spring 2014 for
their helpful feedback: Daryl Deford, Tim Dwyer, Zeb Engberg, Michael Firrisa,
Jeff Hein, Nathan McNew, Jacob Richey, Tom Shemanske, Scott Smedinghoff, and
DavidWebb.
Many thanks go to the others who offered helpful comments and corrections:
France Dacar, Ariyan Javanpeykar, BoGwang Jeon, Chan-Ho Kim, Chao Li, Ben-
jaminLinowitz,NicoleSutherland,JoeQuinn,andJiangweiXue.
I am profoundly grateful to those who offered their encouragement at various
timesduringthewritingofthisbook: SrinathBaba,ChantalDavid,MatthewGreen-
berg and Kristina Loeschner, David Michaels, and my mother Connie Voight. Fi-
nally, Iwouldlike tooffermy deepestgratitude tomy partnerBrianKennedy—this
bookwouldnothavebeenpossiblewithouthispatienceandenduringsupport. Thank
youall!
iv
To do
[[Sections that are incomplete, or comments that need to be followed up on, are
in red.]]
Contents
Contents v
I Algebra 1
1 Introduction 3
1.1 Hamilton’squaternions . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Algebraafterthequaternions . . . . . . . . . . . . . . . . . . . . . 8
1.3 Moderntheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Beginnings 15
2.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Quaternionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 22
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Involutions 27
3.1 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Reducedtraceandreducednorm . . . . . . . . . . . . . . . . . . . 30
3.4 Uniquenessanddegree . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5 Quaternionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 34
3.7 Algorithmicaspects . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Quadraticforms 41
v
vi CONTENTS
4.1 Normform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Nonsingularstandardinvolutions . . . . . . . . . . . . . . . . . . . 46
4.4 Isomorphismclassesofquaternionalgebras . . . . . . . . . . . . . 47
4.5 Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.7 Hilbertsymbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.8 Orthogonalgroups . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.9 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 56
4.10 Algorithmicaspects . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Quaternionalgebrasincharacteristic2 61
5.1 Separabilityandanothersymbol . . . . . . . . . . . . . . . . . . . 61
5.2 Characteristic2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.3 Quadraticformsandcharacteristic2 . . . . . . . . . . . . . . . . . 63
5.4 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 67
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Simplealgebras 69
6.1 The“simplest”algebras . . . . . . . . . . . . . . . . . . . . . . . . 69
6.2 Simplemodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.3 SemisimplemodulesandtheWedderburn–Artintheorem . . . . . . 74
6.4 Centralsimplealgebras . . . . . . . . . . . . . . . . . . . . . . . . 77
6.5 Quaternionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.6 Skolem–Noether . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.7 Reducedtraceandnorm . . . . . . . . . . . . . . . . . . . . . . . 82
6.8 Separablealgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.9 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 84
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7 Simplealgebrasandinvolutions 89
7.1 TheBrauergroupandinvolutions . . . . . . . . . . . . . . . . . . 89
7.2 Biquaternionalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3 Brauergroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.4 Positiveinvolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.5 Endomorphismalgebrasofabelianvarieties . . . . . . . . . . . . . 93
7.6 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 95
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
CONTENTS vii
8 Orders 97
8.1 Integralstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.3 Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.4 Separableorders. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.5 Ordersinamatrixring . . . . . . . . . . . . . . . . . . . . . . . . 101
8.6 Quadraticforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.7 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 103
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
II Algebraicnumbertheory 105
9 TheHurwitzorder 107
9.1 TheHurwitzorder. . . . . . . . . . . . . . . . . . . . . . . . . . . 107
9.2 HurwitzunitsandfinitesubgroupsoftheHamiltonians . . . . . . . 108
9.3 Euclideanalgorithm,sumsoffoursquares . . . . . . . . . . . . . . 110
9.4 Sumsofthreesquares . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.5 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 113
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10 Quaternionalgebrasoverlocalfields 115
10.1 Localquaternionalgebras . . . . . . . . . . . . . . . . . . . . . . . 115
10.2 Localfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.3 Uniquedivisionring,firstproof . . . . . . . . . . . . . . . . . . . 121
10.4 LocalHilbertsymbol . . . . . . . . . . . . . . . . . . . . . . . . . 123
10.5 Uniquedivisionring,secondproof . . . . . . . . . . . . . . . . . . 123
10.6 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
10.7 Splittingfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
10.8 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 130
10.9 Algorithmicaspects . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
11 Latticesandlocalization 133
11.1 Localizationofintegrallattices . . . . . . . . . . . . . . . . . . . . 133
11.2 Localizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
11.3 BitsofcommutativealgebraandDedekinddomains . . . . . . . . . 135
11.4 Latticesandlocalization . . . . . . . . . . . . . . . . . . . . . . . 136
11.5 Adeliccompletions . . . . . . . . . . . . . . . . . . . . . . . . . . 137
11.6 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 137
viii CONTENTS
11.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
12 Discriminants 139
12.1 Discriminantalnotions . . . . . . . . . . . . . . . . . . . . . . . . 139
12.2 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
12.3 Reduceddiscriminant . . . . . . . . . . . . . . . . . . . . . . . . . 143
12.4 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 144
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
13 Quaternionalgebrasoverglobalfields 147
13.1 Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
13.2 Hilbertreciprocityovertherationals . . . . . . . . . . . . . . . . . 149
13.3 Hasse–Minkowskitheoremovertherationals . . . . . . . . . . . . 152
13.4 Globalfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
13.5 Ramificationanddiscriminant . . . . . . . . . . . . . . . . . . . . 157
13.6 Quaternionalgebrasoverglobalfields . . . . . . . . . . . . . . . . 159
13.7 Hasse–Minkowskitheorem . . . . . . . . . . . . . . . . . . . . . . 160
13.8 Representativequaternionalgebras . . . . . . . . . . . . . . . . . . 160
13.9 Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 161
13.10Algorithmicaspects . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
14 QuaternionidealsoverDedekinddomains 163
14.1 Compositionlawsandidealmultiplication . . . . . . . . . . . . . . 163
14.2 Locallyprincipallattices . . . . . . . . . . . . . . . . . . . . . . . 167
14.3 Compatibleandinvertiblelattices . . . . . . . . . . . . . . . . . . 169
14.4 Projective(andproper)modules . . . . . . . . . . . . . . . . . . . 171
14.5 Invertibilitywithastandardinvolution . . . . . . . . . . . . . . . . 173
14.6 Euclideanquaternionorders . . . . . . . . . . . . . . . . . . . . . 176
14.7 Two-sidedideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
14.8 One-sidedideals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
14.9 Minkowskitheory . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
14.10Hermitianforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
14.11Extensionsandfurtherreading . . . . . . . . . . . . . . . . . . . . 182
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
15 QuaternionordersoverDedekinddomains 185
15.1 Classifyingorders . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
15.2 Quadraticmodulesoverrings . . . . . . . . . . . . . . . . . . . . . 189
15.3 Connectionwithternaryquadraticforms . . . . . . . . . . . . . . . 191